PHYSICAL REVIEW C VOLUME 51, NUMBER 5 MAY 1995 Analytic interpretation of universal anharmonic vibrator behavior in the interacting boson approximation model R. V. Jolos, ' P. von Brentano, R. F. Casten, ' and N. V. Zamfir Joint Institute for Nuclear Research, Dubna, Russia Institut fiir Kernphysik, Universitat zu Koln, D 5093-7 Koln, Germany Brookhaven National Laboratory, Upton, New' York 11973 Clark University, Worcester, Massachusetts 01610 Institute of Atomic Physics, Bucharest Ma-gurele, Romania (Received 15 February 1995) The recently discovered nearly universal anharmonic vibrator behavior (with constant anharmonicity) of nuclei with E(4t+)/E(2t+) between 2.0 and 3.15 has been shown to be a natural and nearly automatic outcome of numerical interacting boson approximation (IBA) calculations. Here, we present an approximate analytic derivation and discussion of this based on the idea of the Q phonon. PACS number(s): 21. 10. Re, 21. 60.Fw Recently, it was shown [1, 2] that a nearly universal em- pirical behavior characterizes nuclei between the vibrator and rotor limits. Specifically, for all nuclei from Z=38 — 82 with 2.05~R4/2 — — E(4t+)/E(2t ) ~3. 15, E(4, +) is empirically linear in E(2t ), with a slope of 2.0. Such behavior is described by the equation E(4, +) =2. 0E(2, )+e4, (2) where e4 is the (constant) intercept. This equation is that of an anharmonic vibrator (AHV) and can be generalized for higher spin yrast states as H= Bnd KQ ' Q, (4) AHV equations actually reflects an underlying phonon struc- ture of nearly all collective nonrotational nuclei. An equally surprising theoretical result has also recently been obtained [2]. The interacting boson approximation (IBA) model [4] naturally reproduces the linear AHV behav- ior of both yrast energies and B(E2) values, for a very wide range of Hamiltonian parameters. We show in Fig. 1 (reproduced from Ref. [2]) the empiri- cal and IBA results for a4 to highlight both the remarkable linearity of the data (i.e. , the adequacy of an AHV interpre- tation with constant anharmonicity) and the excellent repro- duction of this behavior with the IBA. The only significant constraint on the IBA calculations of this AHV behavior is that, with the IBA Hamiltonian n(n — 1) n(n — 1)(n — 2) E(n) = nE(2, +) + e4+ e 6, (3) where Q=(std+dts)+y(dtd)I I, (5) where n=I/2 is viewed as the phonon number and where a6 is another parameter. Such an expression has been dis- cussed, for example, by Das, Dreizler, and Klein [3]. For I 8, it is sometimes useful to include the a6 term. a6 is found to be much smaller than e4. The remarkable feature is that the anharmonicity, e4, is constant for nuclei of such varying underlying structure. The AHV behavior is also reflected [2] in yrast B(E2) values as well, raising again the question of whether the success of the the calculated R4/2 value must be less than 3.15 (i. e. , non- rotational nuclei) and the quantity 1 3 e/4tcN) — 1 — — + 2 N 2N' (6) where N is the boson number. Hence e/4ttN~0. 1 to 0.4. A practical upper limit on e/4t&N is — 2. 0 0.0 e4 O. 16 I 0.5 Ez+ (MeV) I 1. 0 eV 0.0 I 0.5 1. 0 Ea. (MeV) FIG. 1. Correlation of E(4, +) with E(2, +). Left, empirical values for all collective, nonrota- tional even-even nuclei (i. e. , 2.05~R+z~3. 15) for Z=38 — 82. Right, IBA calculations for a broad range of parameters that give this same range of R4&z values. Based on Ref. [2]. 0556-2813/95/5 1(5)/2298(4)/$06. 00 51 R2298 1995 The American Physical Society